# Photon trapping in static axially symmetric spacetime

## TL;DR

This paper explores photon trapping and null geodesic structures in static axially symmetric spacetimes, extending the formalism to non-separable geodesic equations and revealing photon regions akin to Kerr black holes.

## Contribution

It introduces a formalism for describing photon regions and trapping surfaces in static axially symmetric spacetimes with non-separable geodesics, generalizing previous methods.

## Key findings

- Photon regions in static axially symmetric spacetimes resemble Kerr photon spheres.
- The formalism describes trapping surfaces in closed form without integrating geodesic equations.
- Photon regions are 'thickened' due to oblateness, indicating complex lensing features.

## Abstract

Recently, several new characteristics have been introduced to describe null geodesic structure of strong gravitational field, such as photon regions, transversely trapping surfaces and some generalizations. They give an alternative and concise way to describe lensing and shadow features of compact objects with strong gravitational field without recurring to complete integration of the geodesic equations. Here we test this construction in the case of the Weyl metrics when geodesic equations are non-separable, and thus can not be integrated analytically, while the above characteristic surfaces and regions can be described in a closed form. We develop further our formalism for a class of static axially symmetric spacetimes introducing more detailed specification of transversely trapping surfaces in terms of their principal curvatures. Surprisingly, we find in the static case without spherical symmetry certain features, such as photon regions, previously known in the Kerr space. These photon regions can be regarded as photon spheres, "thickened" due to oblateness of the metric.